14.02 Problem Set 4Dennis V. Perepelitsa12 April 2006Ruben Segura-CayuelaT/F/U1. True. If capital never deprecated, then every year’s investment in capital will definitelycause the output level to rise. Since the production function exhibits diminishing returns to scalefrom capital, however, the rate at which output rises will decrease, but never actually get to 0.Growth could go on forever if capital never deprecated, but it would be constantly slowing down.2. Uncertain. An increase in savings will not increase the rate of growth of output per capita,but it will raise the maximum level of output per capita. If you save more, the higher the level ofmaximum output per capita your country will have, but you will not help it get there any faster.3. True. If the savings rate in the U.S. is below the level of savings associated with theGolden Rule level of capital, then an increase in the savigs rate will lead to an increase in long-runconsumption and output per worker. If these are desirable phenomena, then individuals shouldincrease their savings.4. False. The convergence hypothesis states that well-develop e d countries will see the growthof their output per capita slow relative to well-developed countries with less output per capita thanthem. This will cause the output per capita of well-developed countries to converge, or come closertogether, over time. The size of the country in question is irrelevant; it is the output per capitathat matters.5. False. Output per capita is equal to the output per worker multiplied by ratio of employ-ment to population. In the United States in 2000, this ratio was closer to 50% than 60%. Thus,output per capita in the U.S. is roughly equal to 50% of output per worker.Short Question Ia. We start with the following three equations that relate the six variables ut, ut−1, πt, πt−1, gyt, gmt.ut− ut−1= −0.4(gyt− 3%) Okun’s Law (1)πt− pit−1= −(ut− 5%) Phillips Curve (2)gyt= gmt− πtAggregate Demand (3)We rearrange (1), and substitute (3) into (1) for gyt:ut= ut−1− 0.4gmt+ 0.4πt+ 1.2%Next, we rearrange (2):πt= 5% + πt−1− utNow we substitute (2) into (1) for πtand (1) into (2) for ut:ut= ut−1− 0.4gmt+ 0.4(5% + πt−1− ut) + 1.2%πt= 5% + πt−1− (ut−1− 0.4gmt+ 0.4πt+ 1.2%)1Rearranging, we get two equation that define (ut, πt) in terms of the variables (ut−1, πt−1), and theconstant gmt.1.4ut= ut−1− 0.4gmt+ 3.2% + 0.4πt−11.4πt= 3.8% + πt−1− ut−1+ 0.4gmtAnd again:ut=11.4ut−1−0.41.4gmt+3.21.4% +0.41.4πt−1πt=3.81.4% +11.4πt−1−11.4ut−1+0.41.4gmtb. If gmtis reduced from 13% to 3%, then we must look at the medium-run equilibrium (uT, πT)- that is, when π and u have stopped changing. In other words, when uT= uT −1and πT= πT −1.Plugging this in, we have, for our two equations:uT=11.4uT−0.41.4(3%) +3.21.4% +0.41.4πTπT=3.81.4% +11.4πT−11.4uT+0.41.4(3%)After some algebra, we have0.4uT= 2.0% + 0.4πT0.4πT= 5.0% − uTWe plug (2) into (1) to solve for uT, and then use this to solve for πT:0.4uT= 2.0% + (5.0% − uT)1.4uT= 7.0%uT= 5.0%0.4πT= 5.0% − uT0.4πT= 5.0% − (5.0%)πT= 0.0%Comparing this w ith (ut=0, πu=0), we see that the unempoyment rate remains unchanged, but thatthe price level has gone decreased one-for-one for the decrease in monetary policy.c. (As a moral principle, I cannot bring myself to use Microsoft Excel to complete this partof the problem set. Instead, I have written a simple Python script to implement the requestedspreadsheet.) Please find the source code attached as problem1.py, and the output of the scriptattached as problem2-output.txt. Here is a table of the variables (ut, πt) as they vary with fromt = 0, . . . , 15, as well as the constants (gmt, gy, un).2t utπtgmtgyun0 5.0 10.0 3 3 51 7.8571428571428577 7.1428571428571441 3 3 52 9.0816326530612272 3.0612244897959204 3 3 53 8.7900874635568549 -0.72886297376093356 3 3 54 7.4989587671803442 -3.2278217409412777 3 3 55 5.862735764859881 -4.0905575058011596 3 3 56 4.4475091160995843 -3.5380666219007435 3 3 57 3.5944874766709196 -2.1325540985716631 3 3 58 3.3867613123158966 -0.51931541088755928 3 3 59 3.6993108199720526 0.78137376914038859 3 3 510 4.2941859483058629 1.4871878208345262 3 3 511 4.9207579118854818 1.5664299089490452 3 3 512 5.3909499110465013 1.1754799979025452 3 3 513 5.6151013644339436 0.56037863346860295 3 3 514 5.599466298443847 -0.03908766497524363 3 3 515 5.417022308895536 -0.45610997387077878 3 3 5To see that numerical analysis approached the theoretical equilibrium values of (ut, πt), we lookat the values of these variables at t = 35, twenty years later and in the long run:t utπtgmtgyun. . . . . . . . . . . . . . . . . .35 5.0136050339025502 0.017479595468026687 3 3 5So we see that limt→∞ut= 5.0% and limt→∞πt= 0%, as we predicted.d. No. As we can see, inflation is all over the map before settling to 0%. Figure 1 is a chart ofinflation over the next forty years after the change in monetary policy.Short Question IIa. If the Russian GDP in 2000 was 7, 305.65 billion rubles, the population was 146.56 million,and the exchange rate to dollars was 28.129 rubles to the dollar, then the Russian GDP per capitain dollars is:7, 305.65x109rubles146.56x106people1 dollar28.129 rubles= $1, 772.10 per capitab. If the US GDP in 2000 was $9, 816.97 billion, and the population was 284.15 million, thenthe US GDP per capita (in dollars) is:$9, 816.97x109284.15x106people= $34, 548.55 per capitac. Using the exchange rate method, this implies that the Russian GDP is1, 772.109, 816.97= 18% of th e U S GDP.3Figure 1: πt. Inflation versus time.4d. However, we note from http://pwt.econ.upenn.edu that the CGDP relative to the UnitedStates in Russia is 28.06% in the year 2000. Thus, to get a more accurate measurement of therelative GDPs of the country, we divide both values by their CGDP relative to the US (for the USitself, this factor is 100%). So the adjusted Russian GDP is1772.10/.28069, 816.97/1.00= 64% of th e U S GDP.e. Our answers to (c) and (d) differ because we have not taken into account the relative pricelevels in each country. Goods are more expensive in the US - a market basket that costs $28.06 inRussia costs $100.00 in the United States. Thus, we need to account for this in our calculation ofthe relative GDP. We do this by measuring how many market baskets the average GDP will buyin that country, instead of measuring how much those market baskets nominally cost where
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